How to unwrap a specific part of an object with correct size proportions?
Step 1: Have a model with a well-thought-out unwrap.



Step 2: Make changes to one are of the model, breaking the texture coordinates in that area.



Step 3: Select the changed area and re-unwrap it to ensure that all parts have texture coordinates, even if the topology has changed. (I used regular old Unwrap with the Conformal algorithm.) Don't worry too much that it's huge.



Step 4: enable Area distortion display in the UV editor N panel to make it really, really easy to scale the island to the right size. Areas at the right size are displayed with blue.



Step 5: move the re-unwrapped island off to the side. Select the entire model and find any areas that should be directly sewn to the re-unwrapped island. Select these areas and pin those islands. (If you don't have seams because you used Auto Unwrap, use Seams from Islands to get seams.) Select one area (old unwrap + new unwrap) that needs new texture coords and unwrap it with Unwrap. The pinned area will stay in place, while the re-unwrapped area will integrate with the part of the island that already existed.



If you need the UVs to fit into a specific part of a texture, use pinning, live unwrap, snapping, and scaling to 0 along an axis to get them into the right place.

If the areas that changed aren't attached to any existing UV islands, all you need to do is use Area distortion display mode to rescale them until they're blue, then find a spot on the UV map for them.